Quotient set of equivalence class in de Rham cohomology

So the equivalence class [itex]X/\sim[/itex] is the set of all equivalences classes [itex][x][/itex]. I was wondering if there was a way of writing it in terms of the usual quotient operation:
[tex]G/N=\{gN\ |\ g\in G\}?[/tex]

But this appears to me to include no information about the equivalence relation [itex]\sim[/itex], which states that [itex]u[/itex] and [itex]v[/itex] are equivalent if there exists some [itex]w[/itex] such that [itex]u - v = dw[/itex].

Ok, but since each equivalence class can be defined by its specific [itex]dw[/itex] (and [itex]d[/itex] is a unique (one-to-one???) map) then it is defined by its [itex]w[/itex]. Hence it makes more sense that
[tex] H^k = Z^k/B^k = \{z - B^k\ |\ z\in Z^k\}. [/tex]

If [itex]z - b = dw[/itex], then [itex]z= d(w+a)[/itex] since [itex]b=da \in B^k[/itex], which also implies that [itex]z\in B^k[/itex]. But what if there is no [itex]w[/itex] such that [itex]z-b = dw[/itex]? How does this give the rest of the equivalence classes?